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CHAPTER.4 :TRANSISTOR FREQUENCY RESPONSE

• To understand

– Decibels, log scale, general frequency considerations of an amplifier.

– low frequency analysis - Bode plot

– low frequency response – BJT amplifier

– Miller effect capacitance

– high frequency response – BJT amplifier

Introduction

It is required to investigate the frequency effects introduced by the larger capacitive elements

of the network at low frequencies and the smaller capacitive elements of the active device at

high frequencies. Since the analysis will extend through a wide frequency range, the

logarithmic scale will be used.

Logarithms

To say that logaM = x means exactly the same thing as saying ax = M .

For example: What is log28?

• "To what power should 2 be raised in order to get 8?"

• Since 8 is 23 the answer is "3."

• So log28 = 3

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Natural Logarithm (or base e)

• There is another logarithm that is also useful (and in fact more common in natural

processes). Many natural phenomenon are seen to exhibit changes that are either exponentially

decaying (radioactive decay for instance) or exponentially increasing (population growth for

example).

• These exponentially changing functions are written as ea, where „a‟ represents the rate of

the exponential change.

• In such cases where exponential changes are involved, we usually use another kind of

logarithm called natural logarithm. The natural log can be thought of as Logarithm Base-e.

• This logarithm is labeled with ln (for "natural log"), where, e = 2.178.

Semi – Log graph

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Decibels

• The term decibel has its origin in the fact that the power and audio levels are related on a

logarithmic basis. The term bel is derived from the surname of Alexander Graham Bell.

• Bel is defined by the following equation relating two power levels, P1 and P2: G = [log10

P2 / P1] bel

• It was found that, the Bel was too large a unit of measurement for the practical

purposes, so the decibel (dB) is defined such that 10 decibels = 1 bel.

• Therefore,

GdB = [10 log10 P2 / P1 ] dB

• The decimal rating is a measure of the difference in magnitude between two power

levels.

• For a specified output power P2, there must be a reference power level P1.

The reference level is generally accepted to be 1mW.

GdBm = [10 log10 P2 / 1mW ] dBm

GdB = [10 log10 P2 / P1 ] dB

= [10 log10 (V22 / Ri ) / (V12 / Ri )] dB

= 10 l log10 (V2 / V1)2

GdB = [20 log10 V2 / V1 ] dB

One of the advantages of the logarithmic relationship is the manner in which it can be

applied to cascaded stages wherein the overall voltage gain of a cascaded system is the

sum of individual gains in dB.

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Problem1:

AV = (Av1)(Av2)(Av3)…….

AVdB = (Av1dB)+(Av2dB)+(Av3dB)…….

Find the magnitude gain corresponding to a voltage gain of 100dB.

GdB = [20 log10 V2 / V1 ] dB = 100dB

= 20 log10 V2 / V1 ; V2 / V1 = 105 = 100,000

Problem 2:

The input power to a device is 10,000W at a voltage of 1000V. The output power is

500W and the output impedance is 20Ω.

• Find the power gain in decibels.

• Find the voltage gain in decibels.

GdB = 10 log10 (Po/Pi)

= 10 log10 (500/10k)

= -13.01dB

GV = 20 log10 (Vo/Vi)

= 20 log10 (√PR/1000)

= 20 log10 [√(500)(20)/1000]

= - 20dB

( Note: P = V2/R; V = √PR)

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Problem 3 :

An amplifier rated at 40 W output is connected to a 10Ω speaker.

a. Calculate the input power required for full power output if the power gain is 25dB.

b. Calculate the input voltage for rated output if the amplifier voltage gain is 40dB.

a. 25 = 10 log10 40/Pi

Pi = 40 / antilog(2.5) = 126.5mW

b. GV = 20log10Vo/Vi ;

40 = 20log10Vo/Vi

Vo /Vi = antilog 2 = 100

Also, Vo = √PR = √ (40)(10) = 20V Thus, Vi = Vo / 100 = 20/100 = 200mV

General Frequency considerations

• At low frequencies the coupling and bypass capacitors can no longer be replaced by the

short – circuit approximation because of the increase in reactance of these elements.

• The frequency – dependent parameters of the small signal equivalent circuits and the

stray capacitive elements associated with the active device and the network will limit the high

frequency response of the system.

• An increase in the number of stages of a cascaded system will also limit both the high

and low frequency response.

• The horizontal scale of frequency response curve is a logarithmic scale to permit a plot

extending from the low to the high frequency

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• For the RC coupled amplifier, the drop at low frequencies is due to the increasing

reactance of CC and CE, whereas its upper frequency limit is determined by either the parasitic

capacitive elements of the network or the frequency dependence of the gain of the active device.

• In the frequency response, there is a band of frequencies in which the magnitude of the

gain is either equal or relatively close to the midband value.

• To fix the frequency boundaries of relatively high gain, 0.707AVmid is chosen to be the

gain at the cutoff levels.

• The corresponding frequencies f1 and f2 are generally called corner, cutoff, band, break,

or half – power frequencies.

• The multiplier 0.707 is chosen because at this level the output power is half the midband

power output, that is, at mid frequencies,

•

PO mid = | Vo2| / Ro = | AVmidVi|2 / RO

• And at the half – power frequencies,

POHPF = | 0.707 AVmidVi|2 / Ro

= 0.5| AVmid Vi|2 / Ro

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• And, POHPF = 0.5 POmid

• The bandwidth of each system is determined by f2 – f1

• A decibel plot can be obtained by applying the equation,

(AV / AVmid )dB

= 20 log10 (AV / AVmid)

• Most amplifiers introduce a 180° phase shift between input and output signals. At low

frequencies, there is a phase shift such that Vo lags Vi by an increased angle. At high

frequencies, the phase shift drops below 180°.

Low – frequency analysis – Bode plot

In the low frequency region of the single – stage BJT amplifier, it is the RC combinations formed

by the network capacitors CC and CE, the network resistive parameters that determine the cutoff

frequencies.

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Frequency analysis of an RC network

• Analysis of the above circuit indicates that, XC = 1/2πfC ≅ 0 Ω

• Thus, Vo = Vi at high frequencies.

• At f = 0 Hz, XC = ∞, Vo = 0V.

• Between the two extremes, the ratio, AV = Vo / Vi will vary.

As frequency increases, the capacitive reactance decreases and more of the input

voltage appears across the output terminals.

The output and input voltages are related by the voltage – divider rule:

Vo = RVi / ( R – jXC)

the magnitude of Vo = RVi / √R2 + XC2

• For the special case where XC = R, Vo =RVi / R√2 = (1/√2) Vi

AV = Vo / Vi = (1/√2) = 0.707

• The frequency at which this occurs is determined from, XC = 1/2πf1C = R

where, f1 = 1/ 2πRC

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• Gain equation is written as,

AV = Vo / Vi

= R / (R – jXC) = 1/ ( 1 – j(1/ωCR)

= 1 / [ 1 – j(f1 / f)]

• In the magnitude and phase form, AV = Vo / Vi

= [1 /√ 1 + (f1/f)2 ] ∠ tan-1 (f1 / f)

• In the logarithmic form, the gain in dB is

AV = Vo / Vi = [1 /√ 1 + (f1/f)2 ]

= 20 log 10 [1 /√ 1 + (f1/f)2 ]

= - 20 log 10 √ [ 1 + (f1/f)2]

= - 10 log10 [1 + (f1/f)2]

• For frequencies where f << f1 or (f1/ f)2 the equation can be approximated by

AV (dB) = - 10 log10 [ (f1 / f)2]

= - 20 log10 [ (f1 / f)] at f << f1

• At f = f1 ; f1 / f = 1 and

– 20 log101 = 0 dB

At f = ½ f1; f1 / f = 2

– 20 log102 = - 6 dB

• At f = ¼ f1;

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• At f = 1/10 f1;

f1 / f = 4

– 20 log102 = - 12 dB

f1 / f = 10

– 20 log1010 = - 20dB

• The above points can be plotted which forms the Bode – plot.

• Note that, these results in a straight line when plotted in a logarithmic scale.

Although the above calculation shows at f = f1, gain is 3dB, we know that f1 is that

frequency at which the gain falls by 3dB. Taking this point, the plot differs from the straight line

and gradually approaches to 0dB by f = 10f1.

Observations from the above calculations:

• When there is an octave change in frequency from f1 / 2 to f1, there exists corresponding

change in gain by 6dB.

• When there is an decade change in frequency from f1 / 10 to f1, there exists

corresponding change in gain by 20 dB.

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Low frequency response – BJT amplifier

• A voltage divider BJT bias configuration with load is considered for this analysis.

• For such a network of voltage divider bias, the capacitors CS, CC and CE will determine

the low frequency response.

Let us consider the effect of each capacitor independently.

CS:

fLs =1/2π (Rs + Ri)Cs

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Ri = R1 || R2 || Bre

• At mid or high frequencies, the reactance of the capacitor will be sufficiently small to

permit a short – circuit approximations for the element.

• The voltage Vi will then be related to Vs by

Vi |mid = VsRi / (Ri+Rs)

• At f = FLS, Vi = 70.7% of its mid band value.

• The voltage Vi applied to the input of the active device can be calculated using the

voltage divider rule: Effect of CC:

Vi = RiVs / ( Ri+ Rs – jXCs)

• Since the coupling capacitor is normally connected between the output of the active

device and applied load, the RC configuration that determines the low cutoff frequency due to

CC appears as in the figure given below.

• Ro = Rc|| ro

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Effect of CE:

fLC = 1/2n(Ro + RL)Cc

fLE =1/2nReCE

R′s = Rs || R1 || R2

The effect of CE

The effect of CE on the gain is best described in a quantitative manner by recalling that the gain

for the amplifier without bypassing the emitter resistor is given by:

AV = - RC / ( re + RE)

• Maximum gain is obviously available where RE is 0Ω.

• At low frequencies, with the bypass capacitor CE in its “open circuit” equivalent

state, all of RE appears in the gain equation above, resulting in minimum gain.

• As the frequency increases, the reactance of the capacitor CE will decrease,

reducing the parallel impedance of RE and CE until the resistor RE is effectively

shorted out by CE.

• The result is a maximum or midband gain determined by AV = - RC / re.

• The input and output coupling capacitors, emitter bypass capacitor will affect only

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the low frequency response.

• At the mid band frequency level, the short circuit equivalents for these capacitors can be

inserted.

• Although each will affect the gain in a similar frequency range, the highest low frequency

cutoff determined by each of the three capacitors will have the greatest impact.

Problem:

Determine the lower cutoff freq. for the network shown using the following parameters:

Cs = 10 F, CE = 20 F, Cc = 1 F Rs = 1kQ, R1= 40kQ, R2 = 10kQ, RE = 2kQ, RC = 4kQ, RL =

2.2kQ,

B = 100, ro = ooQ, Vcc = 20V

Solution:

a. To determine re for the dc conditions, let us check whether βRE > 10R2

Here, βRE = 200kΩ, 10R2 = 100kΩ. The condition is satisfied. Thus approximate

analysis can be carried out to find IE and thus re.

VB = R2VCC / ( R1+R2) = 4V VE = VB – 0.7 = 3.3V

IE = 3.3V / 2kΩ = 1.65mA

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re = 26mV / 1.65mA = 15.76 Ω

Mid band gain:

AV = Vo / Vi = -RC||RL / re = - 90

• Input impedance

Zi = R1 || R2|| βre = 1.32K

• Cut off frequency due to input coupling capacitor ( fLs)

fLs = 1/ [2π(Rs +Ri)CC1 = 6.86Hz. fLc = 1 / [2π(RC + RL) CC

= 1 / [ 6.28 (4kΩ + 2.2kΩ)1uF]

= 25.68 Hz

Effect of CE:

R′S = RS||R1||R2 = 0.889Ω

Re = RE || (R′S/β + re) = 24.35 Ω

fLe = 1/2π ReCE = 327 Hz

fLe = 327 Hz fLC = 25.68Hz

fLs = 6.86Hz

In this case, fLe is the lower cutoff frequency.

• In the high frequency region, the capacitive elements of importance are the inter-

electrode ( between terminals) capacitances internal to the active device and the wiring

capacitance between leads of the network.

• The large capacitors of the network that controlled the low frequency response are all

replaced by their short circuit equivalent due to their very low reactance level.

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• For inverting amplifiers, the input and output capacitance is increased by a

capacitance level sensitive to the inter-electrode capacitance between the input and output

terminals of the device and the gain of the amplifier.

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Miller Effect Capacitance

• Any P-N junction can develop capacitance. This was mentioned in the chapter on diodes.

• In a BJT amplifier this capacitance becomes noticeable between: the Base- Collector

junction at high frequencies in CE BJT amplifier configurations.

• It is called the Miller Capacitance.

• It effects the input and output circuits.

• Ii = I1 + I2 Eqn (1)

• Using Ohm‟s law yields

I1 = Vi / Zi, I1 = Vi / R1

and I2 = (Vi – Vo) / Xcf

= ( Vi – AvVi) / Xcf

I2 = Vi(1 – Av) / Xcf

Substituting for Ii, I1 and I2 in eqn(1),

Vi / Zi = Vi / Ri + [(1 – Av)Vi] /Xcf

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1/ Zi = 1/Ri + [(1 – Av)] /Xcf

1/ Zi = 1/Ri + 1/ [Xcf / (1 – Av)]

1/ Zi = 1/Ri + 1/ XCM

Where, XCM = [Xcf / (1 – Av)]

= 1/[ω (1 – Av) Cf] CMi = (1 – Av) Cf

CMi is the Miller effect capacitance.

• For any inverting amplifier, the input capacitance will be increased by a Miller effect

capacitance sensitive to the gain of the amplifier and the inter-electrode ( parasitic) capacitance

between the input and output terminals of the active device.

Miller Output Capacitance (CMo)

CMo≅ Cf

Applying KCL at the output node results in:

Io = I1+I2

I1 = Vo/Ro

and I2 = (Vo – Vi) / XCf

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The resistance Ro is usually sufficiently large to permit ignoring the first term of the equation,

thus

Io ≅ (Vo – Vi) / XCf

Substituting Vi = Vo / AV,

Io = (Vo – Vo/Av) / XCf

= Vo ( 1 – 1/AV) / XCf Io / Vo = (1 – 1/AV) / XCf Vo / Io = XCf / (1 – 1/AV)

= 1 / ωCf (1 – 1/AV)

= 1/ ωCMo

CMo = ( 1 – 1/AV)Cf

CMo ≅ Cf [ |AV| >>1]

If the gain (Av) is considerably greater than 1:

CMo≅ Cf

High frequency response – BJT Amplifier

• At the high – frequency end, there are two factors that define the – 3dB cutoff point:

– The network capacitance ( parasitic and introduced) and

– the frequency dependence of hfe(β)

Network parameters

• In the high frequency region, the RC network of the amplifier has the configuration

shown below.

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• At increasing frequencies, the reactance XC will decrease in magnitude, resulting in a

short effect across the output and a decreased gain.

Vo = Vi(-jXC) / R -jXC

Vo / Vi = 1/[ 1+j(R/XC)] ; XC = 1/2πfC AV = 1/[ 1+j(2πfRC)];

AV = 1/[ 1+jf/f2]

o This results in a magnitude plot that drops off at 6dB / octave with increasing frequency.

Network with the capacitors that affect the high frequency response

• Capacitances that will affect the high-frequency response:

• Cbe, Cbc, Cce – internal capacitances

• Cwi, Cwo – wiring capacitances

• CS, CC – coupling capacitors

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• CE – bypass capacitor

The capacitors CS, CC, and CE are absent in the high frequency equivalent of the BJT

amplifier.The capacitance Ci includes the input wiring capacitance, the transition capacitance

Cbe, and the Miller capacitance CMi.The capacitance Co includes the output wiring capacitance

Cwo, the parasitic capacitance Cce, and the output Miller capacitance CMo.In general, the

capacitance Cbe is the largest of the parasitic capacitances, with Cce the smallest.

As per the equivalent circuit,

fH = 1 / 2πRthiCi

Rthi = Rs|| R1||R2||Ri

Ci = Cwi+Cbe+CMi = CWi + Cbe+(1- AV) Cbe

At very high frequencies, the effect of Ci is to reduce the total impedance of the parallel

combination of R1, R2, Ri, and Ci.The result is a reduced level of voltage across Ci, a reduction

in Ib and the gain of the system.

For the output network,

fHo = 1/(2πRThoCo) RTho = RC||RL||ro

Co = Cwo+Cce+CMo

At very high frequencies, the capacitive reactance of Co will decrease and consequently reduce

the total impedance of the output parallel branches.

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The net result is that Vo will also decline toward zero as the reactance Xc becomes smaller.The

frequencies fHi and fHo will each define a -6dB/octave asymtote.

If the parasitic capacitors were the only elements to determine the high – cutoff frequency, the

lowest frequency would be the determining factor.However, the

decrease in hfe(or β) with frequency must also be considered as to whether its break

frequency is lower than fHi or fHo.

hfe (or β) variation

• The variation of hfe( or β) with frequency will approach the following relationship hfe =

hfe mid / [1+(f/fβ)]

• fβ is that frequency at which hfe of the transistor falls by 3dB with respect to its mid band

value.

• The quantity fβ is determined by a set of parameters employed in the hybrid π

model.

• In the hybrid π model, rb includes the

• base contact resistance

• base bulk resistance

• base spreading resistance

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Hybrid π model

• The resistance ru(rbc) is a result of the fact that the base current is somewhat sensitive to

the collector – to – base voltage.

• Since the base – to – emitter voltage is linearly related to the base current through

Ohm‟s law and the output voltage is equal to the difference between the base the base – to –

emitter voltage and collector – to – base voltage, we can say that the

base current is sensitive to the changes in output voltage.

• Thus,

fβ = 1/[2πrπ(Cπ+Cu)]

rπ = βre = hfe mid re

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• Therefore,

fβ = 1/[2π hfemid re(Cπ+Cu)] OR

fβ = 1/[2π βmid re(Cπ+Cu)]

• The above equation shows that, fβ is a function of the bias configuration.

• As the frequency of operation increases, hfe will drop off from its mid band value with a

6dB / octave slope.

• Common base configuration displays improved high frequency characteristics over the

common – emitter configuration.

• Miller effect capacitance is absent in the Common base configuration due to non

inverting characteristics.

• A quantity called the gain – bandwidth product is defined for the transistor by the

condition,

• So that,

| hfemid / [1+j(f/fβ)| = 1

|hfe|dB = 20 log10 | hfemid / [1+j(f/fβ)|

= 20 log101 = 0 dB

• The frequency at which |hfe|dB = 0 dB is indicated by fT.

| hfemid / [1+j(f/fβ)| = 1

hfemid /√ 1+ (fT/fβ)2 ≅ hfemid / (fT/fβ) =1

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( by considering fT>>fβ)

• Thus, fT = hfemid fβ OR fT = βmid fβ

• But, fβ = 1/[2π βmid re(Cπ+Cu)]

fT = (βmid) 1/[2π βmid re(Cπ+Cu)]

fT = 1/[2π re(Cπ+Cu)]

Problem:

For the amplifier with voltage divider bias, the following parameters are given: RS = 1k Ω, R1 =

40kΩ, R2 = 10kΩ, Rc = 4kΩ, RL = 10kΩ

Cs = 10µF, Cc = 1 µF, CE= 20 µF

β = 100, ro = ∞ Ω, VCC = 10Ω

Cπ = 36pF, Cu = 4pF, Cce=1pF, Cwi=6pF, Cwo=8pF

a. Determine fHi and fHo

b. Find fβ and fT

Solution:

To find re, DC analysis has to be performed to find IE.

VB = R2VCC / R1+R2 = 2V VE = 2 – 0.7 = 1.3V

IE = 1.3/1.2K = 1.083mA

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re = 26mV / 1.083mA

re = 24.01Ω,

βre = 2.4kΩ

Ri = RS||R1||R2||βre

Ri = 1.85kΩ

AV = Vo/Vi = - (Rc ||RL) / re

AV = - 119

RThi = Rs||R1||R2||Ri

RThi = 0.6kΩ

To determine fHi and fHo:

fHi = 1/[2πRThiCi] ;

Ci = Cwi+Cbe+(1 – AV)Cbc

= 6pF + 36pF + (1 – (-119)) 4pF

Ci = 522pF

fHi = 1/2πRThiCi fHi = 508.16kHz RTho = Rc||RL

RTho = 2.86kΩ

Co = Cwo+Cce+C Mo

= 8pF+1pF+(1 – (1/-119))4pF

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Co = 13.03pF

fHo = 1/2πRThoCo

fHo = 8.542MHz

fβ = 1/[2π βmid re(Cπ+Cu)]

fβ = 1.66MHz

fT = βfβ

fT = 165.72MHz

Summary – Frequency response of BJT Amplifiers

• Logarithm of a number gives the power to which the base must be brought to obtain the

same number

• Since the decibel rating of any equipment is a comparison between levels, a reference

level must be selected for each area of application.

• For Audio system, reference level is 1mW

• The dB gain of a cascaded system is the sum of dB gains of each stage.

• It is the capacitive elements of a network that determine the bandwidth of a system.

• The larger capacitive elements of the design determine the lower cutoff frequencies.

• Smaller parasitic capacitors determine the high cutoff frequencies.

• The frequencies at which the gain drops to 70.7% of the mid band value are called

– cutoff, corner, band, break or half power frequencies.

• The narrower the bandwidth, the smaller is the range of frequencies that will permit a

transfer of power to the load that is atleast 50% of the midband level.

• A change in frequency by a factor of 2, is equivalent to one octave which results in a 6dB

change in gain.

• For a 10:1 change in frequency is equivalent to one decade results in a 20dB

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change in gain.

• For any inverting amplifier, the input capacitance will be increased by a Miller effect

capacitance determined by the gain of the amplifier and the inter electrode ( parasitic)

capacitance between the input and output terminals of the active device.

CMi = (1 – AV)Cf

• Also, CMo ≅ Cf (if AV >>1)

• A 3dB drop in β will occur at a frequency defined by fβ, that is sensitive to the DC

operating conditions of the transistor.

• This variation in β defines the upper cutoff frequency of the design.

Problems:

1. The total decibel gain of a 3 stage system is 120dB. Determine the dB gain of each stage, if

the second stage has twice the decibel gain of the first and the third has 2.7 times decibel gain of

the first. Also, determine the voltage gain of the each stage.

• Given: GdBT = 120dB

We have GdBT = GdB1+GdB2+GdB3

Given, GdB2 = 2GdB1

GdB3 = 2.7GdB1

Therefore, 120dB = 5.7GdB1

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GdB1 = 21.05, GdB2 = 42.10

GdB3 =56.84

We have GdB = 10 log[Vo / Vi]

Vo / Vi = antilog ( GdB/10) G1 = 127.35

G2 = 16.21k

G3 = 483.05k

Problem 2:

2. If the applied ac power to a system is 5µW at 100mV and the output power is 48W, determine

a. The power gain in decibels b. The output voltage

c. The voltage gain in decibels, if the output impedance is 40kΩ.

d. The input impedance

Given: Pi = 5µW.Vi = 100mV, Po = 48w Ro = 40kΩ

a. GdB =10 log [48/ 5µ] = 69.82

2b. Po = Vo/Ro,

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Vo = √PoRo = 1385.64V

c. Voltage gain in dB = 20 log [1385.64/100m] = 82.83 d. Ri = Vi2 / Pi = 2kΩ

General steps to solve a given problem:

Normally, the amplifier circuit with all the values of biasing resistors, value of β and values inter

electrode capacitances ( Cbe, Cbc and Cce) will be given.

It is required to calculate: fLS, fLC and fLE

Also, fHi, fHo, fβ and fT

• Step1: Perform DC analysis and find the value of IE, and re

– Find the value of Ri ( Zi) using the value of re

– Find the value of AVmid

• Step 2: Find fLS using the formula 1/2π(Ri+RS)CS

• Step 3: Find fLC using the formula 1/2π(RC+RL)CC

• Step 4: Determine the value of fLE using the formula 1/2πReCE

where, Re = RE || [(R′S)/β + re]

R′S = RS||R1||R2

• Step 5: Determine fHi using the formula 1/2πRThiCi

where RThi = R1||R2||RS||βre

Ci = Cwi + Cbe + (1-AV)Cbc

• Step 6: Determine fHo using the formula 1/2πRThoCo

where RTho = RC||RL||ro

Co = Cwo + Cce+ C bc

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• Step 7: Determine fβ using the formula 1/[2π βmid re(Cπ+Cu)]

• Step 8: Determine fT using the formula fT = βmid fβ

Problem: Determine the following for the given network:

1. fLs 2. fLc 3. fLE 4.fHi 5. fHo 6. fβ and fT

• Given:

VCC = 20V, RB = 470kΩ, RC = 3kΩ, RE = 0.91kΩ, RS = 0.6kΩ, RL = 4.7kΩ

CS = CC = 1µF, CE = 6.8 µF

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Cwi = 7pF, Cwo=11pF, Cbe = 6pF, Cbe = 20pF and Cce = 10pF

Solution:

IB = (VCC – VBE) / [RB + (β +1)RE]

IB = 3.434mA

IE = β IB

IE = 3.434mA

re = 26mV / IE

re = 7.56Ω

AV = - (RC||RL) / re

AV = -242.2

Zi = RB|| βre

Zi = 754.78 Ω

fLS = 1/2π(Ri+RS)CS

fLS = 117.47Hz

fLC = 1/2π(RC+RL)CC

fLC = 20.66 Hz

fLE = 1/2πReCE ;

where, Re = [(R′S /β)+ re] || RE

R′S = RB || RS

fLE = 1.752kHz

Ci = Cwi + Cbe + (1 – AV) Cbc

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Ci = 1.48nF

RThi = RS || RB|| βre

RThi = 334.27Ω

fHi = 1 / 2π(1.48nF)(334.37)

fHi = 321.70 KHz

Co = CWo + Cce + (1 – 1/AV) Cbc

Co = 27.02pF

RTho = RC || RL

RTho = 1.83K

fHo = 1 / 2π(27.02p)(1.83k)

fHo = 3.21MHz

fβ = 1 / 2π (100) (7.56)( 20p + 6p)

fβ = 8.09MHz

fT = βfβ

fT = 803MHz